Ensembles and the Postulate of Equal a Priori Probabilities

Liouville's theorem left equilibrium underdetermined: any density that is a function of the conserved quantities is stationary, but mechanics alone does not say which one nature realizes. Statistical mechanics closes the gap with a single assumption about probabilities, from which the entire equilibrium theory follows. This lesson states that assumption, defines the ensembles it generates, and establishes the property that makes the whole scheme predictive: for a system of $10^{22}$ particles the probability distribution over macroscopic variables is so sharply peaked that the average and the most probable value are, for every practical purpose, the same number.¹

Ensembles

A macroscopic measurement does not resolve the microstate. It fixes a few macroscopic constraints — the energy, volume, and particle number of an isolated gas, say — and the system is free to be in any microstate consistent with them. The ensemble formalizes this ignorance.

Gibbs introduced the device: rather than track one system in time, imagine a cloud of identically constrained systems filling phase space with density $\rho$, and compute averages over the cloud at one instant. Three ensembles, matched to three ways of controlling a system, carry almost all of equilibrium statistical mechanics. They differ in which macroscopic quantities are held rigidly fixed and which are allowed to fluctuate through contact with a reservoir.

The microcanonical ensemble is the starting point, because it applies to the isolated system whose constraints are purely mechanical, and because the other two are derived from it by putting a small system in contact with a large microcanonical reservoir. Its construction requires only one physical postulate.

The postulate of equal a priori probabilities

An isolated system has a definite energy (within a tolerance $\delta E$), and its microstate lies somewhere on the energy shell built in the previous lesson. The accessible microstates are those the system can occupy consistent with all its constraints — the microstates in the shell. The fundamental postulate assigns them equal probability.

Three observations fix its status.

• It is consistent with the dynamics. A uniform density on the energy shell is a function of $H$ alone, so by Liouville's theorem it is stationary. The postulate selects, among the infinitely many stationary densities, the one that is flat on the accessible region.
• It is the least-biased assignment. With no information beyond the constraints, assigning equal probability introduces no distinction between microstates that the constraints do not distinguish. This is the maximum-entropy reading, made precise in the next lesson.
• It is a postulate, not a theorem. Attempts to derive it from mechanics (via ergodicity, below) succeed only under assumptions no less strong. Its justification is ultimately that the thermodynamics it produces is correct.

Everything downstream — temperature, entropy, the Boltzmann and Gibbs distributions, the whole machinery of the following modules — is a consequence of this one statement applied to systems in contact.

The accessible volume and macrostate probabilities

The postulate turns questions about equilibrium into questions about counting phase-space volume. Suppose the system has an internal parameter $x$ — the energy in one half of a partitioned box, the number of particles on one side, the total magnetization — that is not fixed by the external constraints. The probability that $x$ takes a given value is proportional to the phase-space volume compatible with it.

The equilibrium macrostate is the most probable one, and the second law is the statement that an isolated system, released from a constrained value of $x$, moves to the value of largest $\Omega$. What makes this a law rather than a tendency is the extraordinary sharpness of $P(x)$ when $N$ is macroscopic.

Counting states: the cell size

The volume $\Omega (E)$ carries the dimensions of $(\text{position}\times \text{momentum}{)}^{3N}$, so it is not yet a pure number of states. Probabilities formed as ratios $\Omega (E,x)/\Omega (E)$ are dimensionless and insensitive to the issue, but the entropy $k\ln \Omega$ and any absolute count of microstates require a unit of phase-space volume. Classical mechanics does not supply one; quantum mechanics does. The uncertainty principle $\Delta q\Delta p\gtrsim h$ forbids localizing a single degree of freedom into a phase-space area smaller than Planck's constant, so each of the $3N$ conjugate pairs occupies a cell of area $h$, and one microstate fills a phase-space volume $h^{3N}$.

Neither factor changes the location of the maximum of $\Omega (E,x)$ or the width of $P(x)$ — both are properties of ratios, in which the constants cancel. They matter for the absolute entropy: $h^{3N}$ sets the zero from which entropy is measured, and the $N!$ restores extensivity, without which the entropy of a gas would depend spuriously on how its particles are labelled. The consequences of the $N!$ — the Sackur-Tetrode entropy and the resolution of the Gibbs paradox — are worked out in the microcanonical and classical-gas modules. Here it is enough that a finite unit of phase-space volume converts Liouville's continuous measure into a countable $\Omega$, and that the counting inherits the sharpness derived next.

Sharpness of the distribution

The multiplicity $\Omega (E,x)$ is a product of the multiplicities of independent degrees of freedom, so its logarithm is a sum of $\mathcal{O}(N)$ terms and is itself extensive, of order $N$. Expand $\ln \Omega (E,x)$ about its maximum at $x=x^{\ast }$:

with no linear term at the maximum. Because $\ln \Omega \sim N$, the second derivative is of order $N$ (write it $-N/{\sigma }_0^2$ for a constant ${\sigma }_0$ set by the microscopic physics). Exponentiating gives a Gaussian,

whose standard deviation is

The relative width — the spread as a fraction of the value, when $x$ is extensive and $x^{\ast }\sim N$ — is

For $N\sim 10^{22}$ this is of order $10^{-11}$: the internal variable is pinned to its most probable value to eleven significant figures. Fluctuations exist, but they are utterly negligible on a macroscopic scale, and this is why thermodynamics reports sharp values for quantities that are, microscopically, statistical. The distinction between the mean $⟨x⟩$, the most probable $x^{\ast }$, and a single measurement dissolves in the thermodynamic limit.

Time averages and the ergodic hypothesis

A real measurement is taken on one system over a stretch of time, not on an imagined cloud of copies at one instant. The quantity a physical apparatus reports is the time average

taken along the single trajectory the system actually follows. The ensemble formalism instead computes the ensemble average $⟨A⟩$ over the microcanonical density. The two agree only if the trajectory visits the energy shell in a way that samples every region in proportion to its phase-space volume.

A single trajectory cannot literally pass through every point of a $(6N-1)$-dimensional surface (it is a one-dimensional curve), so the strict statement is that the trajectory comes arbitrarily close to every point and samples the shell uniformly in the limit. Proving this for a realistic Hamiltonian is extraordinarily hard, and it is false for integrable systems, which possess enough conserved quantities to confine the motion to a low-dimensional torus rather than the full shell. For the many-body systems of interest — gases and liquids with generic interactions — the hypothesis is taken as an empirically successful assumption, on the same footing as the equal-probability postulate itself. The two figures below contrast the object the apparatus samples with the object the ensemble computes.

The route to thermodynamics

The postulate and its sharpness deliver the program of the next lesson. Because $P(x)$ is a Gaussian of relative width $1/\sqrt{N}$, its logarithm is dominated by the single term $\ln \Omega (E,x^{\ast })$, and the equilibrium of an isolated system is the maximum of $\ln \Omega$. Identifying $k\ln \Omega$ with the entropy turns maximize the number of accessible microstates into maximize the entropy, and the conditions for the maximum, worked out when two systems share energy, produce temperature and the second law. The counting introduced here becomes the statistical entropy next.